As discussed in a previous document, the ion distribution function in the thermal-barrier region of a tandem mirror machine can be unstable to electrostatic and/or electromagnetic two-stream modes. In this document the author examines the analytic stability in an infinite medium of those electronic ion-ion two-stream modes which have their wave vector, k, parallel to the axial magnetic field (this is equivalent to examining 1-d localized stability). The point of our doing inhomogenous simulations is precisely to examine any non-infinite medium effects on the instability in a thermal barrier. Any unstable ion-ion two-stream modes are expected to have wavelengths of order wavelength di; since the thermal-barrier cell itself is much longer that this, our analytic (local) approximation will provide at least a zeroth-order estimate of instability as well as a reference from which to measure any inhomogenity effects recovered later by our particle simulation runs.

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